Energetics

If the hypothetical Vibranium absorbs kinetic energy, it probably does so in its own frame of reference. In this frame, the target of the shield is coming with great speed towards the shield. Upon impact, the shield somehow absorbs all of the target's kinetic energy, reducing the target's relative velocity (in the shield's frame) to zero. What happened from an observer's perspective? The shield collided with the target, somehow caused the target to match the velocity of the shield upon impact. In the observer's frame of reference, it could be that the shield actually imparted kinetic energy to the target.

###Where does the kinetic energy go? Presumably Vibranium heats up or deforms ever-so-slightly when it absorbs kinetic energy. Beyond that, it's comic book physics. :-)

Momentum

###How can momentum be conserved?

It's a textbook example of a purely inelastic ("sticky") collision. The shield is moving, hits its target, and the two move with the same final velocity. The shield would have to have changed its velocity during the impact in order to conserve momentum.

How does the shield inflict damage by absorbing kinetic energy?

When the shield impacts its target and somehow makes target match its own speed, the new speed of the target may be very different from its original speed. This impact does damage to the target because the target potentially experiences very large accelerations. It's the same kind of damage that a punch delivers: the painful part of a punch isn't usually in the impact, it's in the sudden (whiplashing) change of velocity that results from the impact.

If a boxer punches an opponent's teeth and knocks a few teeth out, it's not the impact that knocks them out. The teeth break near their roots, yet the boxer's glove doesn't impact the roots. The impact changes the tooth's velocity, which puts incredible shear stress on the tooth's roots as a result. The tooth, being unable to withstand the shear stress caused by the sudden change in velocity, breaks loose at its weakest point.

Other remarks

It might be easier to think of how it works if you consider the limit of the shield having nearly-infinite mass (but Cap somehow is strong enough to wield it).

Energetics

If the hypothetical Vibranium absorbs kinetic energy, it probably does so in its own frame of reference. In this frame, the target of the shield is coming with great speed towards the shield. Upon impact, the shield somehow absorbs all of the target's kinetic energy, reducing the target's relative velocity (in the shield's frame) to zero. What happened from an observer's perspective? The shield collided with the target, somehow caused the target to match the velocity of the shield upon impact. In the observer's frame of reference, it could be that the shield actually imparted kinetic energy to the target.

Where does the kinetic energy go?

Presumably Vibranium heats up or deforms ever-so-slightly when it absorbs kinetic energy. Beyond that, it's comic book physics. :-)

Momentum

How can momentum be conserved?

It's a textbook example of a purely inelastic ("sticky") collision. The shield is moving, hits its target, and the two move with the same final velocity. The shield would have to have changed its velocity during the impact in order to conserve momentum.

How does the shield inflict damage by absorbing kinetic energy?

When the shield impacts its target and somehow makes target match its own speed, the new speed of the target may be very different from its original speed. This impact does damage to the target because the target potentially experiences very large accelerations. It's the same kind of damage that a punch delivers: the painful part of a punch isn't usually in the impact, it's in the sudden (whiplashing) change of velocity that results from the impact.

If a boxer punches an opponent's teeth and knocks a few teeth out, it's not the impact that knocks them out. The teeth break near their roots, yet the boxer's glove doesn't impact the roots. The impact changes the tooth's velocity, which puts incredible shear stress on the tooth's roots as a result. The tooth, being unable to withstand the shear stress caused by the sudden change in velocity, breaks loose at its weakest point.

Other remarks

It might be easier to think of how it works if you consider the limit of the shield having nearly-infinite mass (but Cap somehow is strong enough to wield it).